The full Schwinger-Dyson tower - for coloured tensor modelsuni... · 2. Grundelemente | 2.1 Das Logo > > > > Mathematics Institute, University of Munster SFB 878 Corfu 17. 19 September

The full Schwinger-Dyson towerfor coloured tensor models

Carlos. I. Perez-Sanchez

> 12

Corporate Design Manual 1.1

> 13

> 11

Logofarben und Farbumgebung des Logos in der Praxis>

Das Logo steht üblicherweise in Pantone

Black 7 bzw. in CMYK (90% Schwarz,

10% Yellow) auf weißem Untergrund.

Diese Definition gilt für sämtliche hochwer-

tigen Printerzeugnisse wie zum Beispiel

Briefbogen Premium, Visitenkarte, Flyer,

Folder, Broschüren, Einladungskarten,

Magazine, auf Fahnen und Universitäts-

fahrzeugen, Schildern etc.

Sollte dieser Kontrast in der Wirkung in Aus-

nahmefällen nicht ausreichend sein, kann

das Logo negativ (Weiß) auf dunklen Farb-

hintergründen platziert werden, die zu

dem verbindlichen Farbspektrum (siehe

Kapitel 2.2 Farben) gehören.

Außerdem besteht die Möglichkeit, das Logo

negativ (Weiß) auf einem in Pantone Black 7

eingefärbten Hintergrund zu positionieren.

Grundsätzlich darf das Logo nicht als Outline-

Variante (siehe nächste Seite) verwendet

werden. Außerdem sollte das Logo weder

mit einem harten bzw. weichen Schatten

hinterlegt werden. „Photoshop-Spielereien“,

wie zum Beispiel die Verwendung von Relief-

Filtern sind ausdrücklich nicht gestattet.

Auf Faxbögen, dem Biefbogen Standard und

dem Kurzbrief wird das Logo in 100% Schwarz

angelegt, um einen Qualitätsverlust bei der

Wiedergabe des Logos auf herkömmlichen

Arbeitsplatzdruckern zu vermeiden.

Logo in Pantone Black 7auf weißem Hintergrund

Logo negativ (Weiß)auf Hintergrund in Pantone Black 7

Logo negativ (Weiß)auf Hintergrund inPantone 315

2. Grundelemente | 2.1 Das Logo

>

>

>

>

Mathematics Institute,University of Munster

SFB 878

Corfu 17. 19 September

COST Training School, Qspace (MP 1405)

QSPACE Summer School 2016

QUANTUM STRUCTURE OF SPACETIMEAND GRAVITY

21 – 28 August, 2016Belgrade, Serbia

Topics and lecturers:

Random matrices Francois David Institute for Theoretical Physics, Paris

3D gravity Catherine Meusburger Friedrich-Alexander University, Erlangen

Geometry with fluxes Peter Schupp Jacobs University, Bremen

Fuzzy spaces and applications Harold Steinacker University of Vienna, Vienna

Spectral triples and related Dmitri Vassilevich Federal University ABC, Sao Paulo

The school consists of lectures accompanied by tutorial sessions, as well as some slots reserved for studentpresentations. The aim is to provide the students with the necessary background to pursue original re-search in these and related topics. Students and postdocs from countries participating in COST QSPACEAction are eligible for financial support.

Organizers:

Paolo Aschieri (Italy) Larisa Jonke (Croatia) Harold Steinacker (Austria)John Barrett (UK) Catharine Meusburger (Germany) Richard Szabo (UK)Maja Buric (Serbia) Vincent Rivasseau (France) Marko Vojinovic (Serbia)

Marija Dimitrijevic Ciric (Serbia) Mairi Sakellariadou (UK)

Contact: Info: Application deadline:

[email protected] http://www.qssg16.ipb.ac.rs May 31 2016

OutlineOutlineNon-perturbative approach to quantum (coloured) tensor fields

ϕ4-interaction Λ0 = 1 Λ = N � 1

I correlation functions G(2k)B

G• : {coloured graphs} function space

I full Ward-Takahashi IdentitiesI Schwinger-Dyson equations (joint work with Raimar Wulkenhaar)

C. I. Perez Sanchez (Math. Munster) Outline 7 July 2 / 19

MotivationMotivationTensor models generalize the random 2D geometry of random matrices(“Quantum Gravity”)

Z =∫∑

topologiesgeometries

D[g]e−SEH [g] ∼∫∑

( )∈ topologiesgeometries

µ

(k

1

2

DD

1

2

k

k

1

)

{1 1 , 2 2 , 3 3

}=

{ }

23

1

2 3

1

3

1

23

1

21

2

3

1

2

3

=

tensor models also useful in AdS2/CFT1 (Gurau-WittenSachdev-Ye–Kitaev-like model; course by V. Rivaseeau)

C. I. Perez Sanchez (Math. Munster) Motivation 7 July 3 / 19


Z =∫∑




µ

(k

1

2

DD

1

2

k

k

1

){

1 1 , 2 2 , 3 3

}=

{ }

23

1

2 3

1

3

1

23

1

21

2

3

1

2

3

=




Z =∫∑




µ

(k

1

2

DD

1

2

k

k

1

){

1 1 , 2 2 , 3 3

}=

{ }

23

1

2 3

1

3

1

23

1

21

2

3

1

2

3

=



Matrix modelsC as “Rank-2 tensor models”

For complex matrix models∫D[M, M]e−Tr(MM†)−λV(M, M†)

M

M

M

M

M

M

M

...

1

= dual triangulation to...

1

.

For V(M, M) = λTr((MM†)2), different connected O(λ2)-graphs are

12

1 2

1 2

12

0

0

0

0

1

12

1 2

1 2

12

0

0

0

0

rectangular matrices, M ∈MN1×N2(C) and M W(1)M(W(2))t

(W(a) ∈ U(Na)). U(N1)×U(N2)-invariants are Tr((MM†)q), q ∈ Z≥1

C. I. Perez Sanchez (Math. Munster) Ribbon graphs as coloured ones 7 July 4 / 19

Matrix modelsC as “Rank-2 tensor models”

For complex matrix models∫D[M, M]e−Tr(MM†)−λV(M, M†)

M

M

M

M

M

M

M

...

1

= dual triangulation to...

1

.

For V(M, M) = λTr((MM†)2), different connected O(λ2)-graphs are

12

1 2

1 2

12

0

0

0

0

1

12

1 2

1 2

12

0

0

0

0

rectangular matrices, M ∈MN1×N2(C) and M W(1)M(W(2))t

(W(a) ∈ U(Na)). U(N1)×U(N2)-invariants are Tr((MM†)q), q ∈ Z≥1

C. I. Perez Sanchez (Math. Munster) Ribbon graphs as coloured ones 7 July 4 / 19

Coloured Tensor ModelsColoured Tensor Models

a QFT for tensors ϕa1 ...aD and ϕa1 ...aD, whose indices transform

independently under each factor of G = U(N1)×U(N2)× . . .×U(ND)

for each g = (W(1), . . . , W(D)) ∈ G, W(a) ∈ U(Na),

ϕa1a2 ...aD

g(ϕ′)a1a2 ...aD = W(1)

a1b1W(2)

a2b2. . . W(D)

aDbDϕb1 ...bD

ϕa1a2 ...aD

g(ϕ′)a1a2 ...aD = W(1)

a1b1W(2)

a2b2. . . W(D)

aDbDϕb1b2 ...bD

S0[ϕ, ϕ] = Tr2(ϕ, ϕ) = is the kinetic term and higher G-invariants

serve as interaction vertices. For instance, the ϕ43-theory has:

Sint[ϕ, ϕ] = λ

(1

+

1

+

1

)

Z =∫D[ϕ, ϕ] e−N2(S0+Sint)[ϕ,ϕ] (with D[ϕ, ϕ] = ∏

a

dϕadϕa2πi

)

C. I. Perez Sanchez (Math. Munster) Coloured Tensors 7 July 5 / 19

An example of O(λ4) Feynman graph of the ϕ43-model, where is the

propagator:

1C. I. Perez Sanchez (Math. Munster) Coloured Tensors 7 July 6 / 19

1

=

1

2

3

1

2

3

2

3

12

3

11

2

3

1

2

3

3

1

23

1

2

0 0

0

0

00

0

0

0 0

0

0

00

0

0

0 0

0

0

00

0

0

1

Vertex bipartite regularly edge–D-coloured graphs

(Vacuum) Feynman graphs of a model V, Feynvac.D (V), are (D + 1)-coloured

graphs. PL-manifolds can be crystallized [Pezzana, ‘74] by such graphs

1/N-expansion

A(G) = λV(G)/2N F(G)−D(D−1)4 V(G)︸︷︷︸

=: D− 2(D−1)! ω(G) generalizes g; not topol. invariant

= exp(−SRegge[N, D, λ])

[Gurau, ’09], [Bonzom, Gurau, Riello, Rivasseau, ’11]


. . .

1

12

1 2

21

2 1

0

0

0

0

1

. . .

a

b

c

d ΓC

C / ∼(Objects mod ∼)

Manifolds GraphsPezzana

Theorem

Hbubble∗

π1

Physical γC

On the low-dim topology of colored tensor models without Pezzana’stheorem [CP 2017, J.Geom.Phys. 120 (2017) (arXiv:1608.00246)]


Correlation functionsCorrelation functionsreplace the expansion of log ZQFT[J] by the respective CTM-one

log ZQFT[J] =∞

∑n=0

1n!

∫dx1 · · ·dxn G(n)

conn.(x1, x2, . . . , xn)J(x1)J(x2) · · · J(xn)

FeynD(V) = {open Feynman diagrams of the (rank-D) tensor model V}.The boundary ∂G of G ∈ FeynD(V) has as vertex-set the external legs of Gand as a-coloured edges 0a-paths in G between them.

2 2

1

12

2

1 1

2

2 1

1

∂V2

1

2

1

2

1

33

3

1

=1

Also

F =2

12

12

1

333

has as boundary1

, but F /∈ Feyn(ϕ43).


log ZQFT[J] =∞

∑n=0

1n!

∫dx1 · · ·dxn G(n)



2 2

1

12

2

1 1

2

2 1

1

∂V2

1

2

1

2

1

33

3

1

=1

Also

F =2

12

12

1

333

has as boundary1



log ZQFT[J] =∞

∑n=0

1n!

∫dx1 · · ·dxn G(n)



2 2

1

12

2

1 1

2

2 1

1

∂V2

1

2

1

2

1

33

3

1

=1

Also F =2

12

12

1

333

has as boundary1


Expansion of the free energy

im ∂V = ∂ FeynD(V) is the boundary sector of the model V

W[J, J] =∞

∑k=1

∑B∈∂FeynD(V(ϕ,ϕ))

2k=#(B(0))

1|Autc(B)|

G(2k)B ? J(B) .

Coloured automorphisms of B

Jx1

Jx2

Jxk

......

Jy1

Jy2

Jyk

G

(J(B))(x1, . . . , xk︸︷︷︸(ZD)k

) = Jx1 · · · Jxk Jy1 · · · Jyk

Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0

F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a

F(a) · J(B)(a)

C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19



W[J, J] =∞

∑k=1

∑B∈ im ∂V

2k=#(Vertices ofB)

1|Autc(B)|

G(2k)B ? J(B) .


Jx1

Jx2

Jxk

......

Jy1

Jy2

Jyk

G

(J(B))(x1, . . . , xk︸︷︷︸(ZD)k

) = Jx1 · · · Jxk Jy1 · · · Jyk


F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a

F(a) · J(B)(a)




W[J, J] =∞

∑k=1


2k=#(B(0))

1|Autc(B)|

G(2k)B ? J(B) .


Jx1

Jx2

Jxk

......

Jy1

Jy2

Jyk

G

(J(B))(x1, . . . , xk︸︷︷︸(ZD)k

) = Jx1 · · · Jxk Jy1 · · · Jyk


F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a

F(a) · J(B)(a)




W[J, J] =∞

∑k=1


2k=#(B(0))

1|Autc(B)|

G(2k)B ? J(B) .

Coloured automorphisms of BJx1

Jx2

Jxk

......

Jy1

Jy2

Jyk

G (J(B))(x1, . . . , xk︸︷︷︸(ZD)k

) = Jx1 · · · Jxk Jy1 · · · Jyk


F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a

F(a) · J(B)(a)




W[J, J] =∞

∑k=1


2k=#(B(0))

1|Autc(B)|

G(2k)B ? J(B) .


Jx2

Jxk

......

Jy1

Jy2

Jyk

G (J(B))(x1, . . . , xk︸︷︷︸(ZD)k

) = Jx1 · · · Jxk Jy1 · · · Jyk


F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a

F(a) · J(B)(a)




W[J, J] =∞

∑k=1


2k=#(B(0))

1|Autc(B)|

G(2k)B ? J(B) .


Jx2

Jxk

......

Jy1

Jy2

Jyk

G (J(B))(x1, . . . , xk︸︷︷︸(ZD)k

) = Jx1 · · · Jxk Jy1 · · · Jyk


F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a

F(a) · J(B)(a)


1

n = 2

n = 4

n = 6

n = 8

n = 10 . . . . .....

G(2)

1

G(4)1 1

G(4)2 2

G(4)3 3

G(4)|

1

|1

|

G(6) G(6)1

G(6)2

G(6)3

G(6)

32

G(6)

31

G(6)

21

. . .

G(8)j iiij

ll

G(8)j il

1

G(8)

i

jG(8)

il lj

G(8)l

j i

G(8) G(8)i

G(8)a

ba

c

ca

b

. . .SDE

SDE

SDE

SDE

WTI

WTI

WTI

WTI

[CP](arXiv:1608.08134) [CP, R. Wulkenhaar](arXiv:1706.07358)


WD=3[J, J]

WD=3[J, J] = G(2)

1

? J(1

) +

12!

G(4)|

1

|1

| ? J(1

t2) +

12 ∑

cG(4)

c c ? J( c )

+13 ∑

cG(6)

c ?

J( c )

+13

G(6)? J( )

+ ∑i

G(6)

i? J(

i

)+

13!

G(6)|

1

|1

|1

| ? J(1

t3) +

12 ∑

cG(6)|

1

| c c | ? J(

1

t c )+

12! · 22 ∑

cG(8)| c c | c c | ? J

( c t c )+

122 ∑

c<iG(8)| c c | i i | ?

J( c t i

)+

14!

G(8)|

1

|1

|1

|1

| ? J(1

t4) +

12 · 2! ∑

cG(8)|

1

|1

| c c | ? J(

1

t1

t c )+

13

G(8)|

1

| | ? J(

1

t)+

13 ∑

cG(8)

|1

|c|? J(

1

tc )

+ ∑i

G(8)

|1

| i |? J(

1

t

i

)+ ∑

j ; l<iG(8)

j il1

? J(

l jliij

j

j i

i

l

l

)+ ∑

j 6=iG(8)

j iiij

ll

? J(

j iiiij

j

j l

l

l

l

)+

14 ∑

jG(8)

j?

J(

j

j

j

j

)+ ∑

j 6=iG(8)

i

j? J(

i

i

j)+ ∑

iG(8)

il lj

? J( il

l i

lj

i

i

jl)+ ∑

l 6=i 6=jG(8)

lj i

? J( ll

l l

j i

i

i

i

)+

G(8)ca

ca

? J( ca

ca

b

b )+ G(8)

a

ba

c

ca

b

? J(

a

b

ba

c

a

b

a)+O(10) .

WD=3[J, J]

WD=3[J, J] = G(2)

1

? J(1

) +12!

G(4)|

1

|1

| ? J(1

t2) +

12 ∑

cG(4)

c c ? J( c )

+13 ∑

cG(6)

c ?

J( c )

+13

G(6)? J( )

+ ∑i

G(6)

i? J(

i

)+

13!

G(6)|

1

|1

|1

| ? J(1

t3) +

12 ∑

cG(6)|

1

| c c | ? J(

1

t c )+

12! · 22 ∑

cG(8)| c c | c c | ? J

( c t c )+

122 ∑

c<iG(8)| c c | i i | ?

J( c t i

)+

14!

G(8)|

1

|1

|1

|1

| ? J(1

t4) +

12 · 2! ∑

cG(8)|

1

|1

| c c | ? J(

1

t1

t c )+

13

G(8)|

1

| | ? J(

1

t)+

13 ∑

cG(8)

|1

|c|? J(

1

tc )

+ ∑i

G(8)

|1

| i |? J(

1

t

i

)+ ∑

j ; l<iG(8)

j il1

? J(

l jliij

j

j i

i

l

l

)+ ∑

j 6=iG(8)

j iiij

ll

? J(

j iiiij

j

j l

l

l

l

)+

14 ∑

jG(8)

j?

J(

j

j

j

j

)+ ∑

j 6=iG(8)

i

j? J(

i

i

j)+ ∑

iG(8)

il lj

? J( il

l i

lj

i

i

jl)+ ∑

l 6=i 6=jG(8)

lj i

? J( ll

l l

j i

i

i

i

)+

G(8)ca

ca

? J( ca

ca

b

b )+ G(8)

a

ba

c

ca

b

? J(

a

b

ba

c

a

b

a)+O(10) .

WD=3[J, J]

WD=3[J, J] = G(2)

1

? J(1

) +12!

G(4)|

1

|1

| ? J(1

t2) +

12 ∑

cG(4)

c c ? J( c )

+13 ∑

cG(6)

c ?

J( c )

+13

G(6)? J( )

+ ∑i

G(6)

i? J(

i

)+

13!

G(6)|

1

|1

|1

| ? J(1

t3) +

12 ∑

cG(6)|

1

| c c | ? J(

1

t c )

+1

2! · 22 ∑c

G(8)| c c | c c | ? J

( c t c )+

122 ∑

c<iG(8)| c c | i i | ?

J( c t i

)+

14!

G(8)|

1

|1

|1

|1

| ? J(1

t4) +

12 · 2! ∑

cG(8)|

1

|1

| c c | ? J(

1

t1

t c )+

13

G(8)|

1

| | ? J(

1

t)+

13 ∑

cG(8)

|1

|c|? J(

1

tc )

+ ∑i

G(8)

|1

| i |? J(

1

t

i

)+ ∑

j ; l<iG(8)

j il1

? J(

l jliij

j

j i

i

l

l

)+ ∑

j 6=iG(8)

j iiij

ll

? J(

j iiiij

j

j l

l

l

l

)+

14 ∑

jG(8)

j?

J(

j

j

j

j

)+ ∑

j 6=iG(8)

i

j? J(

i

i

j)+ ∑

iG(8)

il lj

? J( il

l i

lj

i

i

jl)+ ∑

l 6=i 6=jG(8)

lj i

? J( ll

l l

j i

i

i

i

)+

G(8)ca

ca

? J( ca

ca

b

b )+ G(8)

a

ba

c

ca

b

? J(

a

b

ba

c

a

b

a)+O(10) .

WD=3[J, J]

WD=3[J, J] = G(2)

1

? J(1

) +12!

G(4)|

1

|1

| ? J(1

t2) +

12 ∑

cG(4)

c c ? J( c )

+13 ∑

cG(6)

c ?

J( c )

+13

G(6)? J( )

+ ∑i

G(6)

i? J(

i

)+

13!

G(6)|

1

|1

|1

| ? J(1

t3) +

12 ∑

cG(6)|

1

| c c | ? J(

1

t c )+

12! · 22 ∑

cG(8)| c c | c c | ? J

( c t c )+

122 ∑

c<iG(8)| c c | i i | ?

J( c t i

)+

14!

G(8)|

1

|1

|1

|1

| ? J(1

t4) +

12 · 2! ∑

cG(8)|

1

|1

| c c | ? J(

1

t1

t c )+

13

G(8)|

1

| | ? J(

1

t)+

13 ∑

cG(8)

|1

|c|? J(

1

tc )

+ ∑i

G(8)

|1

| i |? J(

1

t

i

)+ ∑

j ; l<iG(8)

j il1

? J(

l jliij

j

j i

i

l

l

)+ ∑

j 6=iG(8)

j iiij

ll

? J(

j iiiij

j

j l

l

l

l

)+

14 ∑

jG(8)

j?

J(

j

j

j

j

)+ ∑

j 6=iG(8)

i

j? J(

i

i

j)+ ∑

iG(8)

il lj

? J( il

l i

lj

i

i

jl)+ ∑

l 6=i 6=jG(8)

lj i

? J( ll

l l

j i

i

i

i

)+

G(8)ca

ca

? J( ca

ca

b

b )+ G(8)

a

ba

c

ca

b

? J(

a

b

ba

c

a

b

a)+O(10) .

Boundary graphs and bordisms interpretation

for quartic melonic theories, the boundary sector is the set of all (possibledisconnected) coloured graphs

since B = ∂G represents the ‘boundary of a simplicial complex that Gtriangulates’, one can give a bordism-interpretation to the Green’s functions

for instance, if |∆(B)| = S3 t (S2 × S1) t L3,1 =M, then GB = ∂W[J, J]/∂Bdescribes the bulk compatible with the triangulation of M

ϕ4-generated

4D-bulk

S3

L3,1

S2 × S1

+ϕ4-generated

4D-bulk

S3

L3,1

S2 × S1

+ϕ4-generated

4D-bulk

S3

L3,1

S2 × S1

+ . . .

Each correlation function supports also a 1/N-expansion, G(2k)B = ∑

ω

G(2k,ω)B .


Boundary graphs and bordisms interpretation

for quartic melonic theories, the boundary sector is the set of all (possibledisconnected) coloured graphs

since B = ∂G represents the ‘boundary of a simplicial complex that Gtriangulates’, one can give a bordism-interpretation to the Green’s functions

for instance, if |∆(B)| = S3 t (S2 × S1) t L3,1 =M, then GB = ∂W[J, J]/∂Bdescribes the bulk compatible with the triangulation of M

ϕ4-generated

4D-bulk

S3

L3,1

S2 × S1

+ϕ4-generated

4D-bulk

S3

L3,1

S2 × S1

+ϕ4-generated

4D-bulk

S3

L3,1

S2 × S1

+ . . .

Each correlation function supports also a 1/N-expansion, G(2k)B = ∑

ω

G(2k,ω)B .


Theorem (CP) (Full Ward-Takahashi Identity for arbitrary tensor models)

The partition function Z[J, J] of a tensor model with S0 = Tr2(ϕ, Eϕ) such that

Ep1 ...pa−1mapa+1 ...pD − Ep1 ...pa−1napa+1 ...pD = E(ma, na) for each a = 1, . . . , D,

satisfies, as consequence of the U(N) invariance of the path-integral measure,

∑pi∈Z

δ2Z[J, J]δJp1 ...pa−1mapa+1 ...pD δJp1 ...pa−1napa+1 ...pD

−(

δmanaY(a)ma [J, J]

)· Z[J, J]

= ∑pi∈Z

1E(ma, na)

(Jp1 ...ma ...pD

δ

δJ p1 ...na ...pD

− Jp1 ...na ...pD

δ

δJ p1 ...ma ...pD

)Z[J, J]

where

Y(a)ma [J, J] = ∑

Cf(a)C,ma

? J(C) .

S

era

.

.

....

.

.

.

ixξ(r,i,a)

xξ(r,j,a)j

• era7→ .

.

....

C. I. Perez Sanchez (Math. Munster) The WTI for tensor models 7 July 14 / 19

Schwinger-Dyson equationsSchwinger-Dyson equations

SDEs for the ϕ4mel,D-model (k ≥ 2) [CP, R. Wulkenhaar]

Let D ≥ 3 and let B be a connected boundary graph∂G = B ∈ Grphcl

D ⊂ ∂FeynD(ϕ4m,D).

Let X = (x1, . . . , xk) and s = y1. Then G(2k)B obeys

Jx1

Jx2

Jxk

......

Jy1

Jy2

Jyk

G

(1 +

2λ

Es

D

∑a=1

∑qa

G(2)...

1

(sa, qa)

)G(2k)B (X)

=(−2λ)

Es

D

∑a=1

{∑

σ∈Autc(B)σ∗f(a)B,sa

(X) + ∑ρ>1

1E(yρ

a , sa)Z−1

0∂Z[J, J]

∂ςa(B; 1, ρ)(X)

−∑ba

1E(sa, ba)

[G(2k)B (X)−G(2k)

B (X|sa→ba)]}

where (sa, qa) = (q1, q2, . . . , qa−1, sa, qa+1, . . . , qD).

C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 15 / 19

Schwinger-Dyson equationsSchwinger-Dyson equations

SDEs for the ϕ4mel,D-model (k ≥ 2) [CP, R. Wulkenhaar]

Let D ≥ 3 and let B be a connected boundary graph∂G = B ∈ Grphcl

D ⊂ ∂FeynD(ϕ4m,D).

Let X = (x1, . . . , xk) and s = y1. Then G(2k)B obeys

Jx1

Jx2

Jxk

......

Jy1

Jy2

Jyk

G

(1 +

2λ

Es

D

∑a=1

∑qa

G(2)...

1

(sa, qa)

)G(2k)B (X)

=(−2λ)

Es

D

∑a=1

{∑

σ∈Autc(B)σ∗f(a)B,sa

(X) + ∑ρ>1

1E(yρ

a , sa)Z−1

0∂Z[J, J]

∂ςa(B; 1, ρ)(X)

−∑ba

1E(sa, ba)

[G(2k)B (X)−G(2k)

B (X|sa→ba)]}

where (sa, qa) = (q1, q2, . . . , qa−1, sa, qa+1, . . . , qD).


A simple quartic modelA simple quartic model

Proposal of a model with V[ϕ, ϕ] = λ · 1 1

S0[ϕ, ϕ] = Tr2(ϕ, Eϕ) = ∑x∈Z3

ϕx(m2 + |x|2)ϕx , |x|2 = x21 + x2

2 + x23 .

The boundary sector is:

∂ Feyn3( 1 1 ) = {3-coloured graphs with connected components in Θ} ,

being

Θ ={

1

, 1 1 ,1

, 1 , 1 ,1

, . . .}

.

Let X2k be the graph in Θ with 2k vertices, and G(2k) = G(2k)X2k

:

G(2) = G(2)

1

, G(4) = G(4)1 1

, G(6) = G(6)1

, G(8) = G(8)1

, G(10) = G(10)1

.

Full tower of exact equations obtained [CP, R. Wulkenhaar]


The exact 2pt-function equation. Melonic (planar) limit, conjecturally

(m2 + |x|2 + 2λ ∑

q,p∈Z

G(2)mel(x1, q, p)

)·G(2)

mel(x)

= 1 + 2λ ∑q∈Z

1x2

1 − q2

[G(2)

mel(x1, x2, x3)−G(2)mel(q, x2, x3)

]

The exact 2k-pt-function equation. Melonic limit, conjecturally

(1 +

2λ

m2 + |s|2 ∑q,p∈Z

G(2)mel(x

11, q, p)

)·G(2k)

mel (x1, . . . , xk)

=(−2λ)

m2 + |s|2[ k

∑ρ=2

1(xρ

1)2 − (x1

1)2

(G(2ρ−2)

mel (x1, . . . , xρ−1) ·G(2k−2ρ+2)mel (xρ, . . . , xk)

)

− ∑q∈Z

G(2k)mel (x

11, x1

2, x13, x2, . . . , xk)−G(2k)

mel (q, x12, x1

3, x2, . . . , xk)

(x11)

2 − q2

]C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 17 / 19

The exact 2pt-function equation. Melonic (planar) limit, conjecturally

(m2 + |x|2 + 2λ ∑

q,p∈Z

G(2)mel(x1, q, p)

)·G(2)

mel(x)

= 1 + 2λ ∑q∈Z

1x2

1 − q2

[G(2)

mel(x1, x2, x3)−G(2)mel(q, x2, x3)

]

The exact 2k-pt-function equation. Melonic limit, conjecturally

(1 +

2λ

m2 + |s|2 ∑q,p∈Z

G(2)mel(x

11, q, p)

)·G(2k)

mel (x1, . . . , xk)

=(−2λ)

m2 + |s|2[ k

∑ρ=2

1(xρ

1)2 − (x1

1)2

(G(2ρ−2)

mel (x1, . . . , xρ−1) ·G(2k−2ρ+2)mel (xρ, . . . , xk)

)

− ∑q∈Z

G(2k)mel (x

11, x1

2, x13, x2, . . . , xk)−G(2k)

mel (q, x12, x1

3, x2, . . . , xk)

(x11)

2 − q2

]C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 17 / 19

Conclusions & outlookConclusions & outlook(Coloured) tensor field theories [Ben Geloun, Bonzom, Carrozza, Gurau, Krajewski,

Oriti, Ousmane-Samary, Rivasseau, Ryan, Tanasa, Toriumi, Vignes-Tourneret,. . .] providea framework for 3 ≤ D-dimensional random geometry

I A bordism interpretation of the correlation functions was givenI A (non-perturbative) Ward-Takahashi identity [CP] based that for

matrix models has been foundI It has been used to derive the full tower of SDE [CP-Wulkenhaar]

I Closed equations: hope of a solvable theory for the simple 1 1 -model(spherical 3-geometries)

Outlook:

I Apply these techniques SYK-like (Sachdev-Ye-Kitaev) models [Witten]

I Applications to GFTI Gauge fields on colored graphs (random spaces) by representing graphs

on finite spectral triples

C. I. Perez Sanchez (Math. Munster) The WTI for tensor models 7 July 18 / 19

ReferencesReferencesM. Disertori, R. Gurau, J. Magnen, and

V. Rivasseau.Phys. Lett. B, 649 95–102, 2007.arXiv:hep-th/0612251.

Bonzom, V. and Gurau, R. and Riello, A.and Rivasseau, V.Nucl. Phys. B 853, 2011arXiv:1105.3122

R. Gurau,Commun. Math. Phys. 304, 69 (2011)arXiv:0907.2582 [hep-th] .

C. I. Perez-Sanchez, R. WulkenhaararXiv:1706.07358

C. I. Perez-Sanchez.J.Geom.Phys. 120 262-289 (2017) (arXiv:1608.00246)and arXiv:1608.08134

E. WittenarXiv:1610.09758v2 [hep-th].

Thank you for your attention!

Documents

The full Schwinger-Dyson tower - for coloured tensor modelsuni... · 2. Grundelemente | 2.1 Das Logo > > > > Mathematics Institute, University of Munster SFB 878 Corfu 17. 19 September